Combining Philosophers

Ideas for William S. Jevons, Michael Dummett and George Dickie

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4 ideas

4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / c. Basic theorems of ST
10537 The ordered pairs <x,y> can be reduced to the class of sets of the form {{x},{x,y}} [Dummett]
     Full Idea: A classic reduction is the class of ordered pairs <x,y> being reduced to the class of sets of the form {{x},{x,y}}.
     From: Michael Dummett (Frege Philosophy of Language (2nd ed) [1973], Ch.14)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
9193 ZF set theory has variables which range over sets, 'equals' and 'member', and extensionality [Dummett]
     Full Idea: ZF set theory is a first-order axiomatization. Variables range over sets, there are no second-order variables, and primitive predicates are just 'equals' and 'member of'. The axiom of extensionality says sets with the same members are identical.
     From: Michael Dummett (The Philosophy of Mathematics [1998], 7)
     A reaction: If the eleven members of the cricket team are the same as the eleven members of the hockey team, is the cricket team the same as the hockey team? Our cricket team is better than our hockey team, so different predicates apply to them.
9194 The main alternative to ZF is one which includes looser classes as well as sets [Dummett]
     Full Idea: The main alternative to ZF is two-sorted theories, with some variables ranging over classes. Classes have more generous existence assumptions: there is a universal class, containing all sets, and a class containing all ordinals. Classes are not members.
     From: Michael Dummett (The Philosophy of Mathematics [1998], 7.1.1)
     A reaction: My intuition is to prefer strict systems when it comes to logical theories. The whole point is precision. Otherwise we could just think about things, and skip all this difficult symbolic stuff.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
10542 To associate a cardinal with each set, we need the Axiom of Choice to find a representative [Dummett]
     Full Idea: We may suppose that with each set is associated an object as its cardinal number, but we have no systematic way, without appeal to the Axiom of Choice, of selecting a representative set of each cardinality.
     From: Michael Dummett (Frege Philosophy of Language (2nd ed) [1973], Ch.14)